White was both a pitcher and shortstop in high school. In 2006, he went 10-0 with a 0.46 ERA, fanning 136 while allowing only 11 walks and 34 hits in 76 innings. He also hit .506 and slugged 1.106. Baseball America named him as a third-team All-American utility player after Kyle Drabek and Kyler Burke. Due to a strong college commitment, he fell to the 14th round of the 2006 amateur draft, when the Los Angeles Dodgers selected him, but he did not sign with them.

In 2008, White improved to 13-3, 2.83 with 113 K and 78 hits allowed in 101 2/3 IP. He led the Atlantic Coast Conference in wins, was second in opponent batting average (.215), third in strikeouts (4 behind leader Chris Hernandez) and was 4th in ERA. He made first-team All-ACC and was named the ACC Pitcher of the Year. White was named second-team or third-team All-American by the three major sources. He was 5-0 with a 2.76 ERA in the postseason. He won three games in the 2008 College World Series, beating Louisiana State University twice and eventual champion Fresno State once. He was named to the All-Tournament Team as the top right-handed pitcher in the Series.

White had an excellent pro debut with the Kinston Indians on April 8, 2010, with five shutout innings of 2-hit, 7-K ball. He finished that first professional season with a record of 10-10, 2.45 between Kinston and AA Akron. In 150 2/3 innings (the most of any Indians farmhand), he allowed only 123 hits while striking out 117 batters. Baseball America rated him as the #8 prospect in the Eastern League, between Kyle Gibson and Jason Kipnis.

After the season, Baseball America ranked him the #47 prospect in baseball, following a #63 ranking the previous year, before he had made his professional debut. He was assigned to AAA Columbus to start the 2011 season, going 1-0, 1.90 in his first four starts. The Indians then announced they would call him up to the majors to make his first start on April 30, replacing the injured Carlos Carrasco in the rotation. Facing the Detroit Tigers, he did quite well, giving up 2 runs on 6 hits in 6 innings and leaving the game with a no-decision. He then registered his first major league win in his second start, on May 5, pitching 6 innings in a 4-3 win over the Los Angeles Angels. He was 1-0 with a 3.60 ERA after three starts when he sprained a ligament in his middle finger while throwing a slider during a game against the Cincinnati Reds on May 20. He was placed on the disabled list with an expected recovery time of two months. On July 30th, he was one of four young players sent by the Indians to the Colorado Rockies for P Ubaldo Jimenez, alongside Drew Pomeranz, Joe Gardner and Matt McBride. The Rockies inserted him in their starting rotation at the end of August, and he earned his first win for them in his second start, a 5-4 win over the San Diego Padres at PETCO Park on September 3rd. He went 2-4, 8.42 in 7 starts for the Rockies.

In 2012, White began the season at AAA Colorado Springs before being placed in the Rockies' starting rotation in early May. He would alternate between the two cities all year, going 3-4, 3.71 in 11 starts in the Pacific Coast League, and 2-9, 5.51 in 23 games (20 starts) for Colorado. He only pitched 98 big league innings, as the Rockies toyed with a new approach to starting pitching, limiting their starters to 75 innings per start. As Alex was not particularly economical with his pitches (he walked 51 batters during those 98 innings), he could not go deep into games, making it hard for him to pick up wins or accumulate innings. On December 4th, he was traded to the Houston Astros along with minor leaguer Alex Gillingham in return for Wilton Lopez, a reliever known for his outstanding control.

According to the UNC website, Alex's brother Travis White played one year in the pros, but no confirmation has been found